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%I #70 Jan 08 2022 22:03:34
%S 1,2,6,17,50,147,435,1290,3834,11411,34001,101400,302615,903632,
%T 2699598,8068257,24121674,72137547,215786649,645629160,1932081885,
%U 5782851966,17311097568,51828203475,155188936431,464732722872
%N T(n,0) + T(n,1) + ... + T(n,n), T given by A027907.
%C Let b(n)=a(n) mod 2; then b(n)=1/2+(-1)^n*(1/2-A010060(floor(n/2))). - _Benoit Cloitre_, Mar 23 2004
%C Binomial transform of A027306. Inverse binomial transform of = A032443. Hankel transform is {1, 2, 3, 4, ..., n, ...}. - _Philippe Deléham_, Jul 20 2005
%C Sums of rows of the triangle in A111808. - _Reinhard Zumkeller_, Aug 17 2005
%C Number of 3-ary words of length n in which the number of 1's does not exceed the number of 0's. - _David Scambler_, Aug 14 2012
%C The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - _Peter Bala_, Jan 07 2022
%H Reinhard Zumkeller, <a href="/A027914/b027914.txt">Table of n, a(n) for n = 0..1000</a>
%H Moa Apagodu and Ji-Cai Liu, <a href="https://arxiv.org/abs/1907.13547">Congruence properties for the trinomial coefficients</a>, arXiv:1907.13547 [math.NT], 2019.
%F a(n) = ( 3^n + A002426(n) )/2; lim n -> infinity a(n+1)/a(n) = 3; 3^n < 2*a(n) < 3^(n+1). - _Benoit Cloitre_, Sep 28 2002
%F From _Benoit Cloitre_, Jan 26 2003: (Start)
%F a(n) = (1/2) *( Sum{k = 0..n} binomial(n,k)*binomial(n-k,k) + 3^n );
%F a(n) = Sum_{k = 0..n} Sum_{i = 0..k} binomial(n,i)*binomial(n-i,k);
%F a(n) = 3^n/2*(1+c/sqrt(n)+O(n^-1/2)) where c=0.5... (End)
%F c = sqrt(3/Pi)/2 = 0.4886025119... - _Vaclav Kotesovec_, May 07 2016
%F a(n) = n!*Sum(i+j+k=n, 1/(i!*j!*k!)) 0<=i<=n, 0<=k<=j<=n. - _Benoit Cloitre_, Mar 23 2004
%F G.f.: (1+x+sqrt(1-2x-3x^2))/(2(1-2x-3x^2)); a(n) = Sum_{k = 0..n} floor((k+2)/2)*Sum_{i = 0..floor((n-k)/2)} C(n,i)*C(n-i,i+k)* ((k+1)/(i+k+1)). - _Paul Barry_, Sep 23 2005; corrected Jan 20 2008
%F D-finite with recurrence: n*a(n) +(-5*n+4)*a(n-1) +3*(n-2)*a(n-2) +9*(n-2)*a(n-3)=0. - _R. J. Mathar_, Dec 02 2012
%F G.f.: (1+x+1/G(0))/(2*(1-2*x-3*x^2)), where G(k)= 1 + x*(2+3*x)*(4*k+1)/(4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 30 2013
%F From _Peter Bala_, Jul 21 2015: (Start)
%F a(n) = [x^n]( 3*x - 1/(1 - x) )^n.
%F 1 + x*exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + ... is the o.g.f. for A005773. (End)
%F a(n) = (3^n + GegenbauerC(n,-n,-1/2))/2. - _Peter Luschny_, May 12 2016
%p a := n -> simplify((3^n + GegenbauerC(n,-n,-1/2))/2):
%p seq(a(n), n=0..25); # _Peter Luschny_, May 12 2016
%t CoefficientList[ Series[ (1 + x + Sqrt[1 - 2x - 3x^2])/(2 - 4x - 6x^2), {x, 0, 26}], x] (* _Robert G. Wilson v_, Jul 21 2015 *)
%t Table[(3^n + Hypergeometric2F1[1/2 - n/2, -n/2, 1, 4])/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, May 07 2016 *)
%t f[n_] := Plus @@ Take[ CoefficientList[ Sum[x^k, {k, 0, 2}]^n, x], n +1]; Array[f, 26, 0] (* _Robert G. Wilson v_, Jan 30 2017 *)
%o (PARI) a(n)=sum(i=0,n,polcoeff((1+x+x^2)^n,i,x))
%o (PARI) a(n)=sum(i=0,n,sum(j=0,n,sum(k=0,j,if(i+j+k-n,0,(n!/i!/j!/k!)))))
%o (PARI) x='x+O('x^99); Vec((1+x+(1-2*x-3*x^2)^(1/2))/(2*(1-2*x-3*x^2))) \\ _Altug Alkan_, May 12 2016
%o (Haskell)
%o a027914 n = sum $ take (n + 1) $ a027907_row n
%o -- _Reinhard Zumkeller_, Jan 22 2013
%Y Cf. A025191, A027915, A081673, A092255, A055217, A005773.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_
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